Math Explained to Programmers — Householder QR Decomposition

Refactor by Reflection — One API at a Time

Understanding Householder Reflection with Developer Language

Householder QR decomposition elegantly factors matrices into a product of an orthogonal matrix (Q) and an upper triangular matrix (R).

Given a matrix A, the decomposition is:

A = QR

We construct Q as a product of reflection matrices (Hᵢ):

Q = H₁H₂...Hₖ

And R is then obtained by applying these reflections in reverse order to the original matrix A:

R = Hₖ...H₂H₁A

Each Householder matrix Hᵢ is like a refactoring step where you set up a clean interface (an API boundary) to isolate and improve one part without disturbing the whole system.

Unlike Gram-Schmidt (which builds modules incrementally), Householder strategically places “mirrors” (to us developers: clear API boundaries) to restructure your monolithic system into well-separated components.

The Mirror Principle: How Householder Reflections Work

Mathematically, a Householder reflection matrix (H) is defined as:

H = I - 2vvᵀ / (vᵀv)

• v: a vector perpendicular (normal) to the “reflection mirror.”
• I: identity matrix (represents…